A Sampling of Remarkable Groups: Thompson's, Self-similar, Lamplighter, and Baumslag-Solitar (Compact Textbooks in Mathematics)

This textbook offers students with a basic understanding of group theory a preview of several interesting groups they would not typically encounter until later in their academic careers. By presenting these advanced concepts at this stage, they will gain a deeper understanding of the subject and be motivated to explore more of it. Groups covered include Thompson’s groups, self-similar groups, Lamplighter groups, and Baumslag-Solitar groups. Each chapter focuses on one of these groups, and begins by discussing why they are interesting, how they originated, and why they are important mathematically. A collection of specific references for additional reading, topics for further research, and exercises are included at the end of every chapter to encourage students’ continued education. With its accessible presentation and engaging style, __A Sampling of Remarkable Groups__ is suitable for students in upper-level undergraduate or beginning graduate abstract algebra courses. It will also be of interest to researchers in mathematics, computer science, and related fields. Preface 7 Note to Students 7 Note to Instructors 7 Acknowledgments 9 Contents 10 1 Preliminaries 12 1.1 Notation 12 1.2 Basic Terms, Definitions, and Identities 13 1.3 Direct Products and Direct Sums 16 1.4 Free Groups 17 1.5 Group Presentations 19 1.6 Cayley Graphs of Groups 24 1.7 Rooted (n)-ary Trees 29 1.7.1 The Infinite Complete Rooted Binary Tree 31 1.7.2 Automorphisms of Infinite Rooted Trees 32 1.8 Topics for Further Exploration 34 1.8.1 Group Constructions 34 1.8.2 Dehn's Fundamental Problems 38 1.8.3 Amenability 38 1.8.4 Growth and Finitely Generated Groups 39 1.8.5 Dead-End Elements and Dead-End Depth 40 1.9 hndrdcyan55 [Preliminaries]Chapter 1 Exercises 46 2 Thompson's Group F 48 2.1 Introduction 48 2.2 Preliminaries 48 2.3 Using Tree Diagrams to Explore Thompson's Group F 54 2.4 Another Way to Describe Thompson's Group F 55 2.5 Thompson's Group F Presented on an Infinite Generating Set 63 2.6 Topics for Further Exploration 72 2.6.1 (n)-Ary Tree Diagrams 72 2.6.2 Thompson's Groups T and V 72 2.6.3 Amenability 73 2.6.4 Dead-End Depth 74 2.7 hndrdcyan55 [Thompson]Chapter 2 Exercises 74 3 Self-Similar Groups 78 3.1 Introduction 78 3.2 Self-Similarity 79 3.3 Automata 81 3.4 Portraits of Automorphisms 88 3.5 Self-Similar Rules 92 3.6 Schreier Graphs 97 3.7 Examples of Self-Similar Groups 98 3.7.1 The Adding Machine 98 3.7.2 The Grigorchuk's Group 101 3.8 Tower of Hanoi 105 3.9 Contrasts Among Representations 108 3.10 Topics for Further Exploration 110 3.10.1 Reflections on Automata Groups 110 3.10.2 Further Properties of Grigorchuk's Group 111 3.11 hndrdcyan55 [self-similargroups]Chapter 3 Exercises 112 4 The Lamplighter Group L2 116 4.1 Introduction 117 4.2 L2 as a Dynamical System 117 4.3 L2 with Ordered Pair Elements, Using an Infinite Direct Sum 122 4.4 Normal Form for the Lamplighter Group 125 4.5 Length of Elements of L2 127 4.6 L2 Generated by an Automaton 133 4.7 Topics for Further Exploration 137 4.7.1 Alternate Notation for Writing Self-Similar Rules 137 4.7.2 Dead-End Elements of L2 138 4.7.3 Variations on the Lamplighter Group 140 4.8 hndrdcyan55 [Lamplighter]Chapter 4 Exercises 141 5 Baumslag–Solitar Groups 144 5.1 Introduction 144 5.2 Examining Groups Through Their Endomorphisms 145 5.3 Normal Form for Baumslag–Solitar Groups 147 5.4 Non-hopfian Baumslag–Solitar Groups 150 5.5 Some Hopfian Baumslag–Solitar Groups 152 5.5.1 BS(1,n) 152 5.5.2 BS(m,n) 159 5.6 Topics for Further Exploration 161 5.6.1 More on Non-hopfian Groups 161 5.6.2 Further Properties of BS(m, m) 161 5.6.3 Decision Problems, Amenability, and Growth 162 5.7 hndrdcyan55 [bs]Chapter 5 Exercises 163 6 Solutions 164 hndrdcyan55 [Preliminaries]Chapter 1 164 [Thompson]Chapter 2 169 [self-similargroups]Chapter 3 177 [Lamplighter]Chapter 4 183 hndrdcyan55 [bs]Chapter 5 187 References 191 Index 195 Front Matter ....Pages i-xii Preliminaries (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 1-36 Thompson’s Group F (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 37-66 Self-Similar Groups (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 67-104 The Lamplighter Group L2 (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 105-132 Baumslag–Solitar Groups (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 133-152 Solutions (Marianna C. Bonanome, Margaret H. Dean, Judith Putnam Dean)....Pages 153-179 Back Matter ....Pages 181-188